MORPHOLOGICAL WAVELET TRANSFORM WITH ADAPTIVE DYADIC STRUCTURES Zhen James Xiang and Peter J. Ramadge Dept. of Electrical Engineering, Princeton University, Princeton NJ ABSTRACT We propose a two component method for denoising multidi- mensional signals, e.g. images. The first component uses a dynamic programing algorithm of complexity O(N log N ) to find an optimal dyadic tree representation of a given multidi- mensional signal of N samples. The second component takes a signal with given dyadic tree representation and formulates the denoising problem for this signal as a Second Order Cone Program of size O(N ). To solve the overall denoising prob- lem, we apply these two algorithms iteratively to search for a jointly optimal denoised signal and dyadic tree representa- tion. Experiments on images confirm that the approach yields denoised signals with improved PSNR and edge preservation. Index Terms— Wavelet transforms, morphological oper- ations, image enhancement, multidimensional signal process- ing, dynamic programming. 1. INTRODUCTION The morphological wavelet transform [1, 2] is a non-linear wavelet transform that replaces the algebraic operations in the normal wavelet transform with max and min operators. This transform appears to better preserve morphological features (such as edges) in low resolution signals. We have recently shown, [3], that this property is closely connected to the accu- rate estimation of level sets using tree-based complexity regu- larizations [4]. Specifically, for a 1-D function the integral of the level set complexity measure of [4] over all levels yields the L 1 norm of the signal’s Haar morphological wavelet co- efficients. So L 1 regularization of the Haar morphological wavelet coefficients is equivalent to controlling the complex- ity of the function’s level sets. Here we explore the extension of this result to high di- mensional signals and its application to image denoising. A major challenge in accomplishing this extension is that for multidimensional signals the choice of wavelet decomposi- tion structure is not unique. Hence we first extend the result for any fixed dyadic tree signal representation. Then we pose and solve the problem of finding an optimal dyadic tree rep- resentation of a given multidimensional signal. Finally, we propose an iterative algorithm that jointly denoises a signal and finds the dyadic tree representation best adapted to the signal. The paper’s contribution is to provide the theoretical justification for this approach and practical algorithms for its implementation. The paper is organized as follows: After introducing re- quired notation in §2, we establish the main theoretical and algorithmic results in §3. We then provide experiment com- parison with other methods in §4 and conclude in §5. 2. PRELIMINARIES The nodes of a binary tree are indexed by pairs of integers (k,l), k,l ≥ 0. The root has index (0, 0) and node (k,l) has two children with indices (k +1, 2l) and (k +1, 2l + 1). Fix integers d, m ≥ 1. Without loss of generality, we consider signals defined on the domain [0, 1] d . Definition 1. A binary tree T is called a “full dyadic tree” on [0, 1] d with resolution 2 m if it satisfies each of the following: (c1) The root node of T is J 0,0 = [0, 1] d . (c2) Any non-leaf node J k,l has color c ∈{1, 2,...,d}. Its children, J k+1,2l and J k+1,2l+1 , are congruent hyperrectan- gles obtained by cutting J k,l with the d-1 dimensional hyper- plane through the center of J k,l and perpendicular to axis-c. (c3) All leaf nodes are hypercubes of side length 2 -m . By (c3), T has height dm and by (c2), each hyperrectangle at depth k has volume |J k,l | =2 -k (0 ≤ k ≤ dm). Let T be a full dyadic tree on [0, 1] d with resolution 2 m . A signal f : [0, 1] d 7→ R has resolution 2 m if f is constant on every leaf node of T. For any threshold γ consider the level set S γ = {x ∈ [0, 1] d : f (x) ≥ γ }. We can use T as a deci- sion tree to decide if x ∈ S γ since each leaf of T is entirely inside or outside S γ . By recursively merging siblings that are both inside or outside S γ with their parent, we can prune T to T γ , the minimum decision tree required to decide if x ∈ S γ . We let Φ(T γ ) denote a general form of the level set complex- ity regularizer proposed in [4]: Φ(T γ )= ∑ L∈π(Tγ ) φ(|L|) where π(T γ ) is the set of leaves of T γ , and φ(|L|) only de- pends on the size (i.e., the depth) of the leaf L. To make this dependency explicit, let α k = φ(2 -k ) (k =0, 1,...,dm). We assume α 0 =0 and 2α k+1 >α k [3]. We can also compute a morphological Haar wavelet trans- form [2] of f in a bottom-up fashion on the dyadic structure T. We associate leaf nodes J dm,l with the highest resolution signal Ψ dm (l)= f (x)| x∈J dm,l . Then we move up the tree and